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Questions tagged [complex-manifolds]

For questions about complex manifolds.

3
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2answers
49 views

Existence of non-constant holomorphic map between two given compact Riemann surfaces

Given two compact Riemann surfaces $X,Y$, can we always find a non-constant holomorphic map from $X$ to $Y$? In particular, when $Y$ is a elliptic curve, does that map exist? Michael Albanese has ...
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1answer
61 views

Line bundle from $\mathcal{O}(p)\cong \mathcal{O}(q)$

Let $X$ be a compact Riemann surface. If $\mathcal{O}(p)\cong \mathcal{O}(q)$. How to see there exists a line bundle $L$ and $s_1,s_2$ two sections of $L$ such that $s_1$ vanishes only at $p$ and $s_2$...
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0answers
28 views

The holomorphic map from a compact smooth curve $C$ to $\mathbb{C}P^1$ when $H^0(C,\mathcal{O}(p))=2$

Let $C$ be a compact smooth complex curve with $H^0(C,\mathcal{O}(p))=2$. I feel confused with the following words: Denote by $a$ and $b$ two non-collinear sections in $H^0(C,\mathcal{O}(p))$. Then ...
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0answers
60 views

Is $H^0(C,\Omega_X)\cong H^0(C,\Omega_X\otimes\mathcal{O}(-p))$?

If $C$ is a complex curve, then any point is a hypersurface. To a point $p$ in $C$, suppose we have $w(p)=0$ for all $w\in\Omega_X$, then do we have $H^0(C,\Omega_X)\cong H^0(C,\Omega_X\otimes\...
6
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1answer
49 views

An almost complex structure on $M$ is equivalent to a reduction of the structure group of the tangent bundle

Let $M$ be an $2n$-dimensional manifold. Let $\mathcal{F}_{\mathrm{GL}(2n, \mathbb{R})}$ be the frame bundle over $M$. Consider the subgroup $\mathrm{GL}(n, \mathbb{C})\subset\mathrm{GL}(2n, \mathbb{R}...
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1answer
19 views

$\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$ for a global holomorphic p-form $\alpha$

Let $X$ be a compact kahler manifold. And $\alpha\in H^{p,0}(X)$. How to see $\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$?
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1answer
37 views

Induced isomorphism between $H^{1,0}(X)$ and $H^{1,0}(Y)$ implies the induced isomorphism between $H^{0,1}(X)$ and $H^{0,1}(Y)$?

Let $X,Y$ be two compact Kahler manifolds and $f:X\to Y$ is a holomorphic map. If the induced map $f^*:H^1(X)\to H^1(Y)$'s restriction $f^*|_{H^{1,0}}$ an isomorphism from $H^{1,0}(X)\to H^{1,0}(Y)$, ...
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0answers
11 views

Maximum co-dimension of a submanifold given by finitely many holomorphic functions

Let $\Omega$ be a domain in $\mathbb{C}^{m}$ and $f_{1},\ldots,f_{k}:\Omega\mapsto\mathbb{C}$ are holomorphic functions. Assume that the common zero set, $Z(f_{1},\ldots,f_{k})$ is a submanifold in $\...
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1answer
55 views

Compact nowhere dense analytic closed set

For a compact nowhere dense analytic closed set, why is that a finite set? Can we get this set is discrete, so that it's finite? Analytic sets are locally zero sets of holomorphic functions, which ...
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1answer
23 views

Example of a complex manifold with certain qualities.

Does there exist a complex manifold in $n$ complex dimensions which is compact and is also parallellizable? That is, there exists $n$ holomorphic sections who are a basis for the holomorphic tangent ...
3
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1answer
43 views

Why is the matrix representation of an almost complex structure like this?

Let $M$ be a Riemann surface, $J$ be an almost complex structure (i.e. a 1-1 tensor such that $J^2=-I$ and for any $x \in M,v,w \in T_xM, \{v,Jv\}$ is oriented). Consider a conformal coordinate at a ...
11
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1answer
181 views

Does every non-compact Riemann surface embed holomorphically into $\mathbb{C}^2$?

Question: Can every non-compact Riemann surface be holomorphically embedded into $\mathbb{C}^2$? If not, what are some (all?) of the obstructions to such an embedding? This question is partially ...
3
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1answer
47 views

What's going on when we compute $d(\gamma(z)) = \frac{1}{|cz+d|^2}dz$, where $\gamma \in \operatorname{SL}_2(\mathbb Z)$

Let $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb Z)$. Consider the space $\Omega^1(\mathbb H)$ of smooth complex $1$-forms on $\mathbb H$. These ...
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1answer
44 views

Kahler metric and $(1,1)$- forms

I am following this set of lecture notes. On page 3, the author is considering a metric on a complex Kahler manifold of dimension $n$, which is denoted by $g_{\alpha, \bar{\beta}} = \partial_{\alpha}\...
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1answer
36 views

Meromorphic Functions on Riemann Surfaces

My question refers to a step in the proof of Prop. 3.3.5 Szamuely and Tamásin's "Galois groups and fundamental groups": Here the statement and Thm 3.3.3 & lemma 3.3.6: The main ingredients for ...
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0answers
51 views

Unique complex structure on the modular curve $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$

Is the complex structure on the modular curve coming from the quotient $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ unique? (Here $\mathbb{H}$ is the upper half plane in $\mathbb{C}$) According to ...
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0answers
29 views

Can we ignore the ``holomorphic trivialisation'' in the definition of a holomorphic vector bundle?

I have learnt two definitions about holomorphic vector bundles over a complex manifold $M$. $E\to M$ is a smooth complex vector bundle with a trivialisation such that the transition functions are ...
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0answers
27 views

CR holomorphic functions

Let $\Omega \subset \mathbb{C}$ be a domain, $\mathcal{O}(\Omega)$ denote holomorphic functions on $\Omega$ and $\mathcal{C}^{\infty}(\overline{\Omega})$ functions smooth up to the boundary. I'm ...
2
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1answer
73 views

Weierstrass $\wp$-function defines a map from the torus to an elliptic curve. Why is it injective?

For $L$ a lattice in $\mathbb C$, the Weierstrass $\wp$-function is the meromorphic function $$\wp(z) = \frac{1}{z^2} + \sum\limits_{0 \neq \lambda \in L}\frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2}$...
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0answers
46 views

Explanation of complex differential forms in terms of smooth differential forms

This question is a reference request (or a detailed answer if anyone is willing to write one). I want to read a rigorous treatment of complex differential forms for Riemann surfaces (or maybe for ...
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0answers
69 views

Vanishing of the Nijenhuis tensor

The Nijenhuis tensor is defined to be: $$(1):\quad N_J(X,Y)\equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY], $$ for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$: $$(...
2
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1answer
51 views

The pullback line bundle restricted on the exceptional divisor is trivial

Let $\sigma:\hat X\to X$ be the blow up of a point $x\in X$, denote the exceptional divisors $\sigma^{-1}(x)$ by $E$. $L\to X$ is a line bundle. Then we have a pullback line bundle $\sigma^*L\to\hat X$...
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1answer
35 views

Intuitive understanding of complex tori

First part: I want to understand intuitively when two complex structures on a torus agree. Is it true that it all just comes down to the fundamental lattices being similar, i.e., having the same ...
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0answers
146 views

How do I prove this map is a covering Projection

Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...
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0answers
101 views

Is the Calabi-Yau property of a smooth manifold a differential invariant?

In this question I asked if it could happen that two complex manifolds are homeomorphic, and one of them is a Calabi-Yau manifold but the other isn't. It turns out that there are complex surfaces that ...
8
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1answer
155 views

Can a complex manifold that is not a Calabi-Yau manifold be homeomorphic to a Calabi-Yau manifold?

This is a kind of a follow up to this question, which actually already had an answer here, in which it is asserted that Hodge numbers in general are not topological invariants. Could it be so extreme ...
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0answers
49 views

Uniformization for possibly singular analytic curves

Is there a classification of the simply-connected one-dimensional complex analytic spaces?
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0answers
45 views

Manifolds with varying (local or global) dimension

What subject should one look into to understand manifolds with varying dimensions throughout it's structure. For example, Imagine a 2-sphere with a line going through it defining some type of ...
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0answers
25 views

Transition maps for holomorphic vector bundle

In the definition of a holomorphic line bundle of rank $n$, it is required that the transition maps $g_{ij}:U_i\cap U_j\to\text{GL}_n(\Bbb C)$ are holomorphic. What does this actually mean? Are we ...
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0answers
27 views

Is $\mathbb{R}^2$ with a non euclidian metric a Riemann surface?

Provided that a metric g is regular enough (e.g Chern's condition), one can add a conformal structure to $\mathbb{R}^2$ with metric g to make it a Riemann surface (see also this question). But I don'...
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0answers
31 views

Does the locally-ringed spaces viewpoint on topology actually do what we want?

There's a post here about how we know that the morphisms of smooth manifolds as locally-ringed spaces are the same as the morphisms of smooth manifolds as charts and atlases. I didn't understand the ...
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0answers
27 views

Coordinates on riemann surface

I am reading Hitchens text on integral systems A Riemann surface is a one-dimensional complex manifold with a maximal set of coordinate charts $\{U_\alpha,\varphi_\alpha\}_{\alpha\in I}$ where $\...
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0answers
73 views

Definition of holomorphic $\mathbb{C}^{\ast}$-action on a complex manifold

What is the definition of a holomorphic $\mathbb{C}^\ast$-action on a complex manifold $M$? My understanding is that it is a group action of the Lie group $\mathbb{C}^\ast$ (the multiplicative group ...
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0answers
88 views

Compactification of $\mathbb{C}$ and complex curves

I know that one-point compactification of $\mathbb{C}$ gives us Reimann sphere. Also, Riemann sphere is homeomorphic to complex projective line $\mathbb{P}^1(\mathbb{C}) = S^{3}/{\sim}$ where $x \sim \...
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0answers
39 views

Complex structure on $\mathbb{C}P^1$

Let $\mathbb{C}P^1 = \mathbb{C}^2 / \mathbb{C}_{*}$, where $\mathbb{C}_{*} = \mathbb{C} \backslash \{0\}$. Denote by $[z_0, z_1]$ the equivalence class of $(z_0, z_1) \neq (0, 0)$. How can I show ...
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0answers
28 views

Let $M$ be a complex manifold. Why are the bundles $\bigwedge^{p, q}M$ not holomorphic vector bundles over $M$ for $q \neq 0$ ?

Let $M$ be a complex manifold. Can anyone help me understand why $\bigwedge^{p, q}M$ are not holomorphic vector bundles over $M$ for $q \neq 0$ ? I think it suffices to show that the transition maps ...
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0answers
53 views

Extending the $\bar \partial$-Poincaré Lemma by using a partition of unity

Let $U$ be a disc (of possibly infinite radius) in $\mathbb{C}$. Then $U$ has the following property: (P) For every $g \in C^{\infty}_c(U)$, there is an $f \in C^{\infty}_c(U)$ such that $\frac{\...
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1answer
105 views

A continuous homomorphism of analytic Lie groups is analytic?

If $\phi: G_1 \rightarrow G_2$ is a continuous homomorphism of real Lie groups, then $\phi$ is automatically a smooth. Is this also true for complex (analytic) Lie groups? For example, if $G$ is a ...
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1answer
35 views

Is there a complex structure on $\mathbb{R}^2$ such that $f(x,y) = x-iy$ is analytic?

This is an old qual question: Is there a complex structure on $\mathbb{R}^2$ (i.e., an atlas of charts making $\mathbb{R}^2$ a complex analytic manifold) such that $f:\mathbb{R}\to\mathbb{C}$, $f(x,...
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1answer
139 views

Fubini-Study on $\mathbb CP^1$

I am trying to prove that on $\mathbb CP1$ the Fubini-Study form $\omega_{FS}$ is a quarter of the standard form $\omega_{std}= d\theta\wedge dh$ on $S^2$, where we identify $S^2$ with $\mathbb CP^1$ ...
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1answer
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Proving that $\mathbb{P}^{n}(\mathbb{C})$ is homeomorphic to $S^{2n+1}/S^{1}$

$\newcommand{\card}[1]{\lvert{#1}\rvert}$Introduction and outline I define $\mathbb{P}^{n}$ as the quotient of $\mathbb{C}^{n+1}\setminus\{(0,0,\ldots,0)\}$ by the scaling action of $\mathbb{C}^{\...
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0answers
89 views

Complex Vector Bundle vs Holomorphic Vector Bundle vs Almost Complex Structures

I am writing because I am extremely confused with the structure of complex vector bundles. Ok first of all I understand that a complex vector bundle is a just a vector bundle $\pi:E\to X$ such that ...
3
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1answer
39 views

foliation with many tangencies

Suppose you have smooth foliation on a Euclidean ball $\mathbb{B}^{4} \subset \mathbb{C}^{2}$, whose leaves are holomorphic curves with respect to the standard complex structure. Let $(z_{1},z_{2})$ ...
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0answers
81 views

What does being “holomorphic at the cusps” mean?

Let $G$ be a finite index subgroup of $\operatorname{SL}_2(\mathbb{Z})$. Let $f$ be a modular form of a given weight on the upper half plane with respect to $G$. This means that $f$ is a holomorphic ...
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0answers
44 views

Question about the definition of a complex manifold

"If a topological manifold $M$ has even dimension $n=2m$, we can identify $\mathbb{R}^{2m}$ with $\mathbb{C}^m$ and require the transition maps to be complex-analytic; this determines a complex-...
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1answer
86 views

Real Lie group acting on a complex manifold

Let $X$ be a complex manifold and $G$ be a real Lie group acting on $X$ by holomorphic transformations. If $\frak g$ is the Lie algebra of $G$. Suppose $\hat{\mathfrak g}:=\mathfrak g+i\mathfrak g$ ...
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0answers
93 views

Diffeomorphism between two manifolds

I'm reading something about Hopf manifolds. We define the Hopf manifold in the following way. Let $z\in \mathbb{C}$ a non-zero complex number which lies in the open unit disk centered at the origin. ...
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1answer
35 views

Equality of $C^\infty$-functions on a complex manifold

Let $(M,\Omega)$ be a compact Kahler complex manifold of dimension $n$ and let $f,g:M\to \mathbb C$ be two $C^\infty$ functions with the following two propewrties: $\int_M f\Omega=\int_M g\Omega$ $\...
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0answers
136 views

Definition of a modular form in terms of differential forms

I have never understood the definition of a modular form in terms of differential forms. What is formally going on when we rearrange the equation $\frac{gz}{f(z)} = (\frac{d(gz)}{dz})^{-k}$ to the ...
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2answers
78 views

Understanding metrics of Hermitian manifolds

Let $X$ be a complex manifold with a Hermitian metric $h$. Locally, we can consider the space $T^{1,0}X$ with basis $\displaystyle\left\{ \frac{\partial}{\partial z_j} \right\}$ and the space $T^{0,1}...